WE STUDY THE GAIN OF REGULARITY FOR THE INITIAL VALUE PROBLEM FOR A COUPLED NONLINEAR SCHRÖDINGER SYSTEM THAT DESCRIBES SOME PHYSICAL PHENOMENA SUCH AS THE PROPAGATION IN BIREFRINGENT OPTICAL FIBERS, KERR-LIKE PHOTO REFRACTIVE MEDIA IN OPTICS AND BOSE-EINSTEIN CONDENSATES. THIS STUDY IS MOTIVATED BY THE RESULTS OBTAINED BY N. HAYASHI ET AL.

WE ARE CONCERNED WITH THE INITIAL-BOUNDARY PROBLEM ASSOCIATED TO THE KORTEWEG?DE VRIES?KAWAHARA PERTURBED BY A DISPERSIVE TERM WHICH APPEARS IN SEVERAL FLUIDS DYNAMICS PROBLEMS. WE OBTAIN LOCAL SMOOTHING EFFECTS THAT ARE UNIFORM WITH RESPECT TO THE SIZE OF THE INTERVAL. WE ALSO PROPOSE A SIMPLE FINITE DIFFERENT SCHEME FOR THE PROBLEM AND PROVE ITS UNCONDITIONAL STABILITY. FINALLY WE GIVE SOME NUMERICAL EXAMPLES.